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Abstract

Monte Carlo methods are becoming more and more popular in statistics due to the fast development of efficient computing technologies. One of the major beneficiaries of this advent is the field of Bayesian inference. The aim of this thesis is two-fold: (i) to explain the theory justifying the validity of the simulation-based schemes in a Bayesian setting (why they should work) and (ii) to apply them in several different types of data analysis that a statistician has to routinely encounter. In Chapter 1, I introduce key concepts in Bayesian statistics. Then we discuss Monte Carlo Simulation methods in detail. Our particular focus in on, Markov Chain Monte Carlo, one of the most important tools in Bayesian inference. We discussed three different variants of this including Metropolis-Hastings Algorithm, Gibbs Sampling and slice sampler. Each of these techniques is theoretically justified and I also discussed the potential questions one needs too resolve to implement them in real-world settings. In Chapter 2, we present Monte Carlo techniques for the commonly used Gaussian models including univariate, multivariate and mixture models. In Chapter 3, I focused on several variants of regression including linear and generalized linear models involving continuous, categorical and count responses. For all these cases, the required posterior distributions are rigorously derived. I complement the methodological description with analysis of multiple real datasets and provide tables and diagrams to summarize the inference. In the last Chapter, a few additional key aspects of Bayesian modeling are mentioned. In conclusion, this thesis emphasizes on the Monte Carlo Simulation application in Bayesian Statistics. It also shows that the Bayesian Statistics, which treats all unknown parameters as random variables with their distributions, becomes efficient, useful and easy to implement through Monte Carlo simulations in lieu of the difficult numerical/theoretical calculations.